Diane AKA Traska:
M is going to be job hunting soon (it's an amicable split... the place M is working is closing down, and they've known about it for months.) There are two leads and a supervisor that M is comfortable with and has a good working relationship with, and M would like to use them as references. The thing is, this is the first job where such a thing would even be likely to occur (last job was in retail.) How does one go about asking someone if they'd like to be a reference, in a face to face situation? They've got several more weeks working together, so M will be seeing them at work daily, so communication will not be a problem.

baritone108:
M should ask each of them privately if they would mind being listed a a reference on her job applications. M should keep it sort of casual, "A, I am applying for a position with ____ and wondered if it would be o.k. if I listed you as a reference?"

The exception is the supervisor mentioned. If it is M's supervisor M will be required to list that person on the application anyway. If it is not M's supervisor then the above applies.

Diane AKA Traska:
May I assume that since M isn't looking yet, but will be some time after they're no longer working together, a rewording of "A, when I fill out applications for my next job, would it be OK if I listed you as a reference?" would be fine?

Mary Lennox:
If they say yes to being a reference, since their workplace is closing down, remember to get non-work contact details from them!

Arila:
When the time seems right, perhaps right after a meeting discussing the close-down, or whatever, say "Reference, it has been so nice working with you this past <time>. Would you allow me to use you as a reference as I look for new work?" If they say yes, then "Could I have your personal contact info? I think that your phone number and email will do."

I find the companies prefer references from superiors rather than coworkers, and frequently three, so if there was another supervisor who you ever worked for/with, I would also ask them.